Abstract: Noether's theorem tells us that if a system is invariant under a group of symmetries, then we have quantities that are conserved. For example, if a system is invariant under translation, then momentum is conserved. If a system is invariant under rotation, then angular momentum is conserved. One of the challenges in numerical analysis is to make sure that these quantities are still conserved when we solve a system numerically, despite the error inherent in a discrete approximation of a continuous system. Noether's theorem gives us a way to do so: If our numerical method respects the symmetries of the system in an appropriate sense, then our conservation laws will still hold for the numerical approximation.

In the first part of the talk, I'll introduce Lagrangian mechanics and Noether's theorem for ordinary differential equations. Then I'll move on to partial differential equations with Maxwell's equations of electromagnetism. These equations have a gauge symmetry, which via Noether's theorem gives us conservation of charge. If time permits, I'll talk about recent work with Ari on finite element numerical methods for Maxwell's equations that respect this symmetry.